Annexes in affine Coxeter complexes

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their interpretation via folded galleries. We establish general properties of annexes and show that in affine Coxeter groups the annex of any fixed element is finite. In rank-two affine Coxeter complexes, we further describe the geometric structure of annex boundaries using descent sets and configurations of parallel reflections. These results offer a new geometric viewpoint on the structure of the Bruhat order.

💡 Research Summary

The paper introduces a new combinatorial object associated with elements of a Coxeter group, called the “annex”. For a given element x in a Coxeter group W, the annex Annex(x) is defined as the set of all elements y such that x does not lie in the shadow of y, i.e. x ≤ y fails in the Bruhat order. Equivalently, Annex(x) is the complement of the set of elements whose shadows contain x. This definition provides a natural way to study the “upper” Bruhat interval